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We demonstrate that the Carroll limit of general relativity coupled to matter captures the chaotic mixmaster dynamics of near-singularity limits. Zooming in on the behavior of general relativity close to spacelike singularities reveals rich and solvable ultra-local Belinski-Khalatnikov-Lifshitz (BKL) dynamics, which we show to be captured by a Carroll limit. Specifically, building on recent work on geometric Carroll expansions of general relativity, we establish that leading-order Carroll gravity, with suitable matter coupling, accurately describes well-known cosmological billiards behavior. Since the Carroll limit implements the ultra-local limit off shell, this opens up the door to a wide range of possible tractable applications, including spatially inhomogeneous setups and the emergence of spikes at late times. This further suggests that Carroll gravity, along with its subleading corrections, could serve as a valuable tool for studying deep infrared physics in AdS/CFT., Comment: 19 pages; v2: references added

Recent work has shown that introducing higher-curvature terms to the Einstein-Hilbert action causes the approach to a space-like singularity to unfold as a sequence of Kasner eons. Each eon is dominated by emergent physics at an energy scale controlled by higher-curvature terms of a given order, transitioning to higher-order eons as the singularity is approached. The purpose of this paper is twofold. First, we demonstrate that the inclusion of matter dramatically modifies the physics of eons compared to the vacuum case. We illustrate this by considering a family of quasi-topological gravities of arbitrary order minimally coupled to a scalar field. Second, we investigate Kasner eons in the interior of black holes with field theory duals and analyze their imprints on holographic observables. We show that the behavior of the thermal $a$-function, two-point functions of heavy operators, and holographic complexity can capture distinct signatures of the eons, making them promising tools for diagnosing stringy effects near black hole singularities., Comment: 29 pages (including two appendices), 7 figures; v2: references added

We analyze the analytic structure of correlators in the field theory dual to the quantum Ba\~{n}ados-Teitelboim-Zanelli (qBTZ) black hole, a braneworld model incorporating exact backreaction from quantum conformal matter. We first compute the quasi-normal mode (QNM) spectrum of operators with dimension $\Delta$ and spin $s=0,\pm 1/2$. The leading QNMs and their overtones display qualitatively different behavior depending on the branch of qBTZ solution; branch 1 is a conical singularity dressed with a horizon while branch 2 is a quantum-corrected BTZ black hole. Hence the relaxation of probe matter distinguishes the type of singularity cloaked by the horizon. We then turn to pole-skipping locations where Green's functions are not unique. At these points, frequency is proportional to temperature, but momentum exhibits complex temperature dependence due to quantum effects. Under the assumption that the pole-skipping point closest to the origin reflects quantum chaos, we infer the likely behavior of the quantum Lyapunov exponent and butterfly velocity in the dual theory. Finally, we examine pole collisions in complex momentum space, showing that quantum corrections imprint a unique signature on the analytic structure of the poles in retarded Green's functions, resulting in level-crossing phenomena that differ notably from the level-touching phenomena in the uncorrected BTZ geometry., Comment: 70 pages, 27 figures. comments welcome!

We investigate the gravitational dual of a fermionic field theory at finite temperature and charge density in two spatial dimensions, subject to a deformation by a relevant scalar operator. This makes a $(3+1)$-dimensional Einstein-Maxwell system coupled to a free fermion fluid, known as an electron cloud, undergo a holographic renormalization group flow. The inner (Cauchy) horizon is destroyed and the near-singularity metric instead adopts the form of a positive-$p_t$ Kasner cosmology, signaling the collapse of the Einstein-Rosen bridge. Previous studies have suggested that this collapse hinders direct probing of the singularity. Nonetheless, we propose and compute several CFT observables that characterize the interior and near-singularity geometries. These include the thermal $a$-function, which decays with a specific power of $p_t$ as nearly all CFT degrees of freedom are integrated out, and two-point correlators for neutral and charged operators, with the latter directly probing the singularity despite the positive-$p_t$. We also calculate characteristic velocities related to entanglement and complexity growth in the time-evolved thermofield double state, as well as the butterfly effect indicative of operator spreading. Notably, the deformed electron cloud features a Lifshitz IR fixed point and an additional Kasner trans-IR fixed point, absent in neutral RG flows., Comment: 29 pages, 15 figures; v2: additional citations added

We conduct an exhaustive study of the interior geometry of a family of asymptotically AdS$_{d+1}$ hairy black holes in an analytically controllable setup. The black holes are exact solutions to an Einstein-Maxwell-Dilaton theory and include the well-known Gubser-Rocha model. After reviewing the setup, we scrutinize the geometry beyond the horizon, finding that these backgrounds can exhibit timelike or Kasner singularities. We generalize the no inner-horizon theorem for hairy black holes to accommodate these findings. We then consider observables sensitive to the geometry behind the horizon, such as Complexity = Anything and the thermal $a$-function. In the Kasner case, we propose a new variant of complexity that characterizes the late-time rate by the Kasner exponents, extending previous work by J{\o}rstad, Myers and Ruan. Additionally, we elucidate the power-law behavior of the thermal $a$-function near the singularity, directly relating it to the Kasner exponents. Finally, we introduce axion-like fields in the Gubser-Rocha model to study the impact of translational symmetry breaking on the black hole interior. We show that Kasner singularities occur for both explicit and spontaneous symmetry breaking, with the Kasner exponents depending on the strength of broken translations only in the latter case., Comment: v1: 59 pages, 21 figures, v2: references added

Krylov complexity has recently emerged as a new paradigm to characterize quantum chaos in many-body systems. However, which features of Krylov complexity are prerogative of quantum chaotic systems and how they relate to more standard probes, such as spectral statistics or out-of-time-order correlators (OTOCs), remain open questions. Recent insights have revealed that in quantum chaotic systems Krylov state complexity exhibits a distinct peak during time evolution before settling into a well-understood late-time plateau. In this work, we propose that this Krylov complexity peak (KCP) is a hallmark of quantum chaotic systems and suggest that its height could serve as an 'order parameter' for quantum chaos. We demonstrate that the KCP effectively identifies chaotic-integrable transitions in two representative quantum mechanical models at both infinite and finite temperature: the mass-deformed Sachdev-Ye-Kitaev model and the sparse Sachdev-Ye-Kitaev model. Our findings align with established results from spectral statistics and OTOCs, while introducing an operator-independent diagnostic for quantum chaos, offering more 'universal' insights and a deeper understanding of the general properties of quantum chaotic systems., Comment: [v3] Notable updates: (1) Expanded discussion on the sparse Sachdev-Ye-Kitaev model, (2) Improved data accuracy, with additional comments and references included

We consider a truncation of the BMN matrix model to a configuration of two fuzzy spheres, described by two coupled non-linear oscillators dependent on the mass parameter $\mu$. The classical phase diagram of the system generically ($\mu \neq 0$) contains three equilibrium points: two centers and a center-saddle; as $\mu \to 0$ the system exhibits a pitchfork bifurcation. We demonstrate that the system is exactly integrable in quadratures for $\mu=0$, while for very large values of $\mu$, it approaches another integrable point characterized by two harmonic oscillators. The classical phase space is mixed, containing both integrable islands and chaotic regions, as evidenced by the classical Lyapunov spectrum. At the quantum level, we explore indicators of early and late time chaos. The eigenvalue spacing is best described by a Brody distribution, which interpolates between Poisson and Wigner distributions; it dovetails, at the quantum level, the classical results and reemphasizes the notion that the quantum system is mixed. We also study the spectral form factor and the quantum Lyapunov exponent, as defined by out-of-time-ordered correlators. These two indicators of quantum chaos exhibit weak correlations with the Brody distribution. We speculate that the behavior of the system as $\mu \to 0$ dominates the spectral form factor and the quantum Lyapunov exponent, making these indicators of quantum chaos less effective in the context of a mixed phase space., Comment: 34 pages, 3 tables, 21 figures. V2: References added

We review constructions of three-dimensional `quantum' black holes. Such spacetimes arise via holographic braneworlds and are exact solutions to an induced higher-derivative theory of gravity consistently coupled to a large$-c$ quantum field theory with an ultraviolet cutoff, accounting for all orders of semi-classical backreaction. Notably, such quantum-corrected black holes are much larger than the Planck length. We describe the geometry and horizon thermodynamics of a host of asymptotically (anti-) de Sitter and flat quantum black holes. A summary of higher-dimensional extensions is given. We survey multiple applications of quantum black holes and braneworld holography., Comment: Invited review, 97 pages + appendices, multiple figures; v2: references added

We formulate spacetime inequalities applicable to quantum-corrected black holes to all orders of backreaction in semiclassical gravity. Namely, we propose refined versions of the quantum Penrose and reverse isoperimetric inequalities, valid for all known three-dimensional asymptotically anti-de Sitter quantum black holes. Previous proposals of the quantum Penrose inequality apply in higher dimensions but fail when applied in three dimensions beyond the perturbative regime. Our quantum Penrose inequality, valid in three dimensions, holds at all orders of backreaction. This suggests cosmic censorship must exist in non-perturbative semiclassical gravity. Our quantum reverse isoperimetric inequality implies a maximum entropy state for quantum black holes at fixed volume., Comment: 14 pages, 3 figures

Within the framework of extended black hole thermodynamics, where the cosmological constant acts as the thermodynamic pressure and its conjugate as the thermodynamic volume, we analyze the phase structure and thermodynamic geometry of the three-dimensional quantum-corrected BTZ (qBTZ) black hole. Our results uncover two-phase transitions in the $T-S$ plane across all pressures except at a critical value. Numerical analysis reveals continuous critical phenomena along the coexistence curve, with critical exponents of 2 and 3 for the heat capacity at constant pressure and the NTG curvature, respectively. Importantly, these values notably deviate from the well-known critical exponents observed in mean-field Van der Waals (VdW) fluids, where the NTG curvature and heat capacity demonstrate discontinuous criticality. To our knowledge, our investigation is the first exploration of critical behavior in black holes incorporating consistent semiclassical backreaction., Comment: 7 pages, 6 figures. v2: references added

Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity ($C_K$) and Spread complexity ($C_S$) gaining prominence. In this study, we investigate their interplay by considering the complexity of states represented by density matrix operators. After setting up the problem, we analyze a handful of analytical and numerical examples spanning generic two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories, uncovering insightful relationships. For generic pure states, our analysis reveals two key findings: (I) a correspondence between moment-generating functions (of Lanczos coefficients) and survival amplitudes, and (II) an early-time equivalence between $C_K$ and $2C_S$. Furthermore, for maximally entangled pure states, we find that the moment-generating function of $C_K$ becomes the Spectral Form Factor and, at late-times, $C_K$ is simply related to $NC_S$ for $N\geq2$ within the $N$-dimensional Hilbert space. Notably, we confirm that $C_K = 2C_S$ holds across all times when $N=2$. Through the lens of random matrix theories, we also discuss deviations between complexities at intermediate times and highlight subtleties in the averaging approach at the level of the survival amplitude., Comment: v1: 41 pages, 10 figures; v2: references added; v3: matching the published version

Recently, the concept of spread complexity, Krylov complexity for states, has been introduced as a measure of the complexity and chaoticity of quantum systems. In this paper, we study the spread complexity of the thermofield double state within \emph{integrable} systems that exhibit saddle-dominated scrambling. Specifically, we focus on the Lipkin-Meshkov-Glick model and the inverted harmonic oscillator as representative examples of quantum mechanical systems featuring saddle-dominated scrambling. Applying the Lanczos algorithm, our numerical investigation reveals that the spread complexity in these systems exhibits features reminiscent of \emph{chaotic} systems, displaying a distinctive ramp-peak-slope-plateau pattern. Our results indicate that, although spread complexity serves as a valuable probe, accurately diagnosing true quantum chaos generally necessitates additional physical input. We also explore the relationship between spread complexity, the spectral form factor, and the transition probability within the Krylov space. We provide analytical confirmation of our numerical results, validating the Ehrenfest theorem of complexity and identifying a distinct quadratic behavior in the early-time regime of spread complexity., Comment: v1: 27 pages, 17 figures; v2: references added; v3: matching the published version

We study holographic renormalization group (RG) flows perturbed by a shock wave in dimensions $d\geq 2$. The flows are obtained by deforming a holographic conformal field theory with a relevant operator, altering the interior geometry from AdS-Schwarzschild to a more general Kasner universe near the spacelike singularity. We introduce null matter in the form of a shock wave into this geometry and scrutinize its impact on the near-horizon and interior dynamics of the black hole. Using out-of-time-order correlators, we find that the scrambling time increases as we increase the strength of the deformation, whereas the butterfly velocity displays a non-monotonic behavior. We examine other observables that are more sensitive to the black hole interior, such as the thermal $a$-function and the entanglement velocity. Notably, the $a$-function experiences a discontinuous jump across the shock wave, signaling an instantaneous loss of degrees of freedom due to the infalling matter. This jump is interpreted as a `cosmological time skip' which arises from an infinitely boosted length contraction. The entanglement velocity exhibits similar dependence to the butterfly velocity as we vary the strength of the deformation. Lastly, we extend our analyses to a model where the interior geometry undergoes an infinite sequence of bouncing Kasner epochs., Comment: 36 pages, multiple figures. v2: references added. v3: matches JHEP version

We show backreaction of quantum fields on black hole geometries can trigger new thermal phase transitions. Specifically, we study the phase behavior of the three-dimensional quantum-corrected static BTZ black hole, an exact solution to specific semi-classical gravitational equations due to quantum conformal matter, discovered through braneworld holography. Focusing on the canonical ensemble, for large backreaction, we find novel reentrant phase transitions as the temperature monotonically increases, namely, from thermal anti-de Sitter space to the black hole and back to thermal anti-de Sitter. The former phase transition is first-order, a quantum analog of the classical Hawking-Page phase transition, while the latter is zeroth-order and has no classical counterpart., Comment: 5 pages, 4 figures; v2: minor edits, references added, PRD version

We develop a new setting in the framework of braneworld holography to describe a pair of coupled and entangled uniformly accelerated universes. The model consists of two branes embedded into AdS space capping off the UV and IR regions, giving rise to a notion of dS wedge holography. Specializing in a three-dimensional bulk, we show that dS JT gravity can emerge as an effective braneworld theory, provided that fluctuations transverse to the brane are included. We study the holographic entanglement entropy between the branes as well as the holographic complexity within the `complexity=anything' proposal. We reproduce a Page curve with respect to an observer collecting radiation on the UV brane, as long as we take the limit where gravity decouples in that universe, thus acting as a non-gravitating bath. The Page curve emerges due to momentum-space (UV/IR) entanglement and can be understood as analogous to the `confinement-deconfinement' transition in theories with a mass gap. Moreover, the analysis of complexity shows that the hyperfast growth phenomenon is displayed within a set of proposals, while late-time linear growth can be recovered for a different set. Our framework thus provides new test grounds for understanding quantum information concepts in dS space and dS holography., Comment: 30 pages + Appendices; 8 figures. Matches JHEP version

The holographic gauge/gravity duality provides an explicit reduction of quantum field theory (QFT) calculations in the semi-classical large-$N$ limit to sets of `gravitational' differential equations whose analysis can reveal all details of the spectra of thermal QFT correlators. We argue that in certain cases, a complete reconstruction of the spectrum and of the corresponding correlator is possible from only the knowledge of an infinite, discrete set of pole-skipping points traversed by a single (hydrodynamic) mode computed in a series expansion in an inverse number of spacetime dimensions. Conceptually, this reduces the computation of a QFT correlator spectrum to performing a set of purely algebraic manipulations. With the help of the pole-skipping analysis, we also uncover a novel structure underpinning the coefficients that enter the hydrodynamic dispersion relations., Comment: 10 pages, 1 figure

When a hadron is probed at high energy, a nontrivial quantum entanglement entropy inside the hadron emerges due to the lack of complete information about the hadron wave function extracted from this measurement. In the high-energy limit, the hadron becomes a maximally entangled state, with a linear dependence of entanglement entropy on rapidity, as has been found in a recent analysis based on parton description. In this paper, we use an effective conformal field theoretic description of hadrons on the light cone to show that the linear dependence of the entanglement entropy on rapidity found in parton description is a general consequence of approximate conformal invariance and does not depend on the assumption of weak coupling. Our result also provides further evidence for a duality between the parton and string descriptions of hadrons., Comment: 5 pages, 1 figure. v2: Matches published version

A new principle in quantum gravity, dubbed spacetime complexity, states that gravitational physics emerges from spacetime seeking to optimize the computational cost of its quantum dynamics. Thus far, this principle has been realized at the linearized level, in holographic theories with Einstein gravity duals, assuming the so-called 'Complexity-Volume' (CV) proposal. We expand on this proof in two significant directions. First, we derive higher-derivative gravitational equations by including appropriate corrections to the CV dictionary. Second, we show semi-classical equations arise by considering the leading bulk quantum corrections to CV. Our proof is valid for two-dimensional dilaton gravities, where the problem of semi-classical backreaction can be solved exactly. However, we argue the principle should hold more generally, leading us to a concrete proposal for bulk complexity of perturbative excited states in arbitrary dimensions. Our results demonstrate the robustness of spacetime complexity as a guiding principle to understand gravity in terms of quantum computation., Comment: 34 pages (+ Appendices), 2 figures; v2 minor edits, JHEP version

The Ryu-Takayanagi prescription can be cast in terms of a set of microscopic threads that help visualize holographic entanglement in terms of distillation of EPR pairs. While this framework has been exploited for regions with a high degree of symmetry, we take the first steps towards understanding general entangling regions, focusing on AdS$_4$. Inspired by simple constructions achieved for the case of disks and the half-plane, we reformulate bit threads in terms of a magnetic-like field generated by a current flowing through the boundary of the entangling region. The construction is possible for these highly symmetric settings, leading us to a modified Biot-Savart law in curved space that fully characterizes the entanglement structure of the state. For general entangling regions, the prescription breaks down as the corresponding modular Hamiltonians become inherently nonlocal. We develop a formalism for general shape deformations and derive a flow equation that accounts for these effects as a systematic expansion. We solve this equation for a complete set of small deformations and show that the structure of the expansion explicitly codifies the expected nonlocalities. Our findings are consistent with numerical results existing in the literature, and shed light on the fundamental nature of quantum entanglement as a nonlocal phenomenon., Comment: 28 pages, 5 figures; v2: typos corrected and references added; v3: matches JHEP version

Holographic braneworlds are used to present a higher-dimensional origin of extended black hole thermodynamics. In this framework, classical, asymptotically anti-de Sitter black holes map to quantum black holes in one dimension less, with a conformal matter sector that backreacts on the brane geometry. Varying the brane tension alone leads to a dynamical cosmological constant on the brane, and, correspondingly, a variable pressure attributed to the brane black hole. Thus, standard thermodynamics in the bulk, including a work term coming from the brane, induces extended thermodynamics on the brane, exactly, to all orders in the backreaction. A microsopic interpretation of the extended thermodynamics of specific quantum black holes is given via double holography., Comment: 6 pages plus supplementary material, 1 figure; v2: fixed typos in supplementary material, clarifications and references added, PRL version

We construct worldsheet traversable wormholes by considering the effects of a double-trace deformation, $\delta\mathcal{L}\sim h\partial\phi_L\partial\phi_R$, coupling the endpoints of an open string in AdS space. The operator deforming the theory is irrelevant and makes the boundaries bend inward toward the IR. This effect, reminiscent of two-dimensional dilaton gravities, renders the teleportation protocol more efficient and facilitates the transfer of information between the members of the dual Bell pair. We compare our results with those obtained with the standard double-trace deformation, $\delta\mathcal{L}\sim h\phi_L\phi_R$, introduced by Gao, Jafferis and Wall., Comment: 34 pages, multiple figures. v2: minor additions and references updated. Matches JHEP version

Membrane nucleation, a higher dimensional analog of the Schwinger effect, is a useful toy model for vacuum decay. While a non-perturbative effect, the computation of nucleation rates has only been accomplished at weak coupling in the field theory. Here we compute the nucleation rates of spherical membranes using AdS/CFT duality, thus naturally including the effects of strong coupling. More precisely, we consider the nucleation of spherical membranes coupled to an antisymmetric tensor field, a process which renders the vacuum unstable above a critical value of the field strength. We analyze membrane creation in flat and de Sitter space using various foliations of AdS. This is accomplished via instanton methods, where the rate of nucleation is dominated by the semi-classical on-shell Euclidean action. Our findings generalize the holographic Schwinger effect and provide a step toward holographic false vacuum decay mediated by Coleman-De Luccia instantons., Comment: 24 pages + appendices, 5 figures. v2: minor corrections and references added

In three-dimensional de Sitter space classical black holes do not exist, and the Schwarzschild-de Sitter solution instead describes a conical defect with a single cosmological horizon. We argue that the quantum backreaction of conformal fields can generate a black hole horizon, leading to a three-dimensional quantum de Sitter black hole. Its size can be as large as the cosmological horizon in a Nariai-type limit. We show explicitly how these solutions arise using braneworld holography, but also compare to a non-holographic, perturbative analysis of backreaction due to conformally coupled scalar fields in conical de Sitter space. We analyze the thermodynamics of this quantum black hole, revealing it behaves similarly to its classical four-dimensional counterpart, where the generalized entropy replaces the classical Bekenstein-Hawking entropy. We compute entropy deficits due to nucleating the three-dimensional black hole and revisit arguments for a possible matrix model description of dS spacetimes. Finally, we comment on the holographic dual description for dS spacetimes as seen from the braneworld perspective., Comment: 43 pages plus appendices, 12 figures. v2: footnote expanded into an appendix, comments and references added. v3: minor edits, references added, JHEP version

Inspired by the universality of computation, we advocate for a principle of spacetime complexity, where gravity arises as a consequence of spacetime optimizing the computational cost of its own quantum dynamics. This principle is explicitly realized in the context of the Anti-de Sitter/Conformal Field Theory correspondence, where complexity is naturally understood in terms of state preparation via Euclidean path integrals, and Einstein's equations emerge from the laws of quantum complexity. We visualize spacetime complexity using Lorentzian threads which, conceptually, represent the operations needed to prepare a quantum state in a tensor network discretizing spacetime. Thus, spacetime itself evolves via optimized computation., Comment: 9 pages, 2 figures, Received honorable mention for the 2022 Essay Competition of the Gravity Research Foundation; References added, to appear in the October 2022 Special Issue of the International Journal of Modern Physics D

There is some tension between two well-known ideas in holography. On the one hand, subregion duality asserts that the reduced density matrix associated with a limited region of the boundary theory is dual to a correspondingly limited region in the bulk, known as the entanglement wedge. On the other hand, correlators that in the boundary theory can be computed solely with that density matrix are calculated in the bulk via the GKPW or BDHM prescriptions, which require input from beyond the entanglement wedge. We show that this tension is resolved by recognizing that the reduced state is only fully identified when the entanglement wedge is supplemented with a specific infrared boundary action, associated with an end-of-the-world brane. This action is obtained by coarse-graining through a variant of Wilsonian integration, a procedure that we call holographic rememorization, which can also be applied to define other reduced density or transition matrices, as well as more general reduced partition functions. We find an interesting connection with AdS/BCFT, and, in this context, we are led to a simple example of an equivalence between an ensemble of theories and a single theory, as discussed in recent studies of the black hole information problem., Comment: 39+15 pages, 8 figures, v2: references and 2 paragraphs added, typos corrected

The island formula -- an extremization prescription for generalized entropy -- is known to result in a unitary Page curve for the entropy of Hawking radiation. This semi-classical entropy formula has been derived for Jackiw-Teitelboim (JT) gravity coupled to conformal matter using the "replica trick" to evaluate the Euclidean path integral. Alternatively, for eternal Anti-de Sitter black holes, we derive the extremization of generalized entropy from minimizing the microcanonical action of an entanglement wedge. The on-shell action is minus the entropy and arises in the saddle-point approximation of the (nonreplicated) microcanonical path integral. When the black hole is coupled to a bath, islands emerge from maximizing the entropy at fixed energy, consistent with the island formula. Our method applies to JT gravity as well as other two-dimensional dilaton gravity theories., Comment: 5 pages and appendices, 2 figures; minor clarifications added, PRD version

We explore the effect of introducing mild nonlocality into otherwise local, chaotic quantum systems, on the rate of information spreading and associated rates of entanglement generation and operator growth. We consider various forms of nonlocality, both in 1-dimensional spin chain models and in holographic gauge theories, comparing the phenomenology of each. Generically, increasing the level of nonlocality increases the rate of information spreading, but in lattice models we find instances where these rates are slightly suppressed., Comment: 36 pages, 12 figures. V2 includes added citations, credit attributions, and corrected typos

Quantum extremal surfaces (QES), codimension-2 spacelike regions which extremize the generalized entropy of a gravity-matter system, play a key role in the study of the black hole information problem. The thermodynamics of QESs, however, has been largely unexplored, as a proper interpretation requires a detailed understanding of backreaction due to quantum fields. We investigate this problem in semi-classical Jackiw-Teitelboim (JT) gravity, where the spacetime is the eternal two-dimensional Anti-de Sitter ($\text{AdS}_{2}$) black hole, Hawking radiation is described by a conformal field theory with central charge $c$, and backreaction effects may be analyzed exactly. We show the Wald entropy of the semi-classical JT theory entirely encapsulates the generalized entropy - including time-dependent von Neumann entropy contributions - whose extremization leads to a QES lying just outside of the black hole horizon. Consequently, the QES defines a Rindler wedge nested inside the enveloping black hole. We use covariant phase space techniques on a time-reflection symmetric slice to derive a Smarr relation and first law of nested Rindler wedge thermodynamics, regularized using local counterterms, and including semi-classical effects. Moreover, in the microcanonical ensemble the semi-classical first law implies the generalized entropy of the QES is stationary at fixed energy. Thus, the thermodynamics of the nested Rindler wedge is equivalent to thermodynamics of the QES in the microcanonical ensemble., Comment: 101 pages, 5 appendices, 3 figures; JHEP version,except we correct signs in Eqs. (C.27)-(C.32)

Holographic entanglement entropy was recently recast in terms of Riemannian flows or 'bit threads'. We consider the Lorentzian analog to reformulate the 'complexity=volume' conjecture using Lorentzian flows -- timelike vector fields whose minimum flux through a boundary subregion is equal to the volume of the homologous maximal bulk Cauchy slice. By the nesting of Lorentzian flows, holographic complexity is shown to obey a number of properties. Particularly, the rate of complexity is bounded below by conditional complexity, describing a multi-step optimization with intermediate and final target states. We provide multiple explicit geometric realizations of Lorentzian flows in AdS backgrounds, including their time-dependence and behavior near the singularity in a black hole interior. Conceptually, discretized flows are interpreted as Lorentzian threads or 'gatelines'. Upon selecting a reference state, complexity thence counts the minimum number of gatelines needed to prepare a target state described by a tensor network discretizing the maximal volume slice, matching its quantum information theoretic definition. We point out that suboptimal tensor networks are important to fully characterize the state, leading us to propose a refined notion of complexity as an ensemble average. The bulk symplectic potential provides a specific 'canonical' thread configuration characterizing perturbations around arbitrary CFT states. Consistency of this solution requires the bulk satisfy the linearized Einstein's equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating for a principle of 'spacetime complexity'. Lastly, we argue Lorentzian threads provide a notion of emergent time. This article is an expanded and detailed version of [arXiv:2105.12735], including several new results., Comment: 95 pages+appendices, 25 pretty figures. JHEP version

We study a two-dimensional theory of gravity coupled to matter that is relevant to describe holographic properties of black holes with a single rotational parameter in five dimensions (with or without cosmological constant). We focus on the near-horizon geometry of the near-extremal black hole, where the effective theory reduces to Jackiw-Teitelboim (JT) gravity coupled to a massive scalar field. We compute the corrections to correlation functions due to cubic interactions present in this theory. A novel feature is that these corrections do not have a definite sign: for AdS$_5$ black holes the sign depends on the mass of the extremal solution. We discuss possible interpretations of these corrections from a gravitational and holographic perspective. We also quantify the imprint of the JT sector on the UV region, i.e. how these degrees of freedom, characteristic for the near-horizon region, influence the asymptotically far region of the black hole. This gives an interesting insight on how to interpret the IR modes in the context of their UV completion, which depends on the environment that contains the black hole., Comment: 45 pages, 2 figures, v3: references added, minor edits

The continuous min flow-max cut principle is used to reformulate the 'complexity=volume' conjecture using Lorentzian flows -- divergenceless norm-bounded timelike vector fields whose minimum flux through a boundary subregion is equal to the volume of the homologous maximal bulk Cauchy slice. The nesting property is used to show the rate of complexity is bounded below by "conditional complexity", describing a multi-step optimization with intermediate and final target states. Conceptually, discretized Lorentzian flows are interpreted in terms of threads or gatelines such that complexity is equal to the minimum number of gatelines used to prepare a CFT state by an optimal tensor network (TN) discretizing the state. We propose a refined measure of complexity, capturing the role of suboptimal TNs, as an ensemble average. The bulk symplectic potential provides a 'canonical' thread configuration characterizing perturbations around arbitrary CFT states. Its consistency requires the bulk to obey linearized Einstein's equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating a notion of 'spacetime complexity'., Comment: 5 pages + Supplemental Material. v3: version accepted for publication in PRL

Quantum corrections to holographic entanglement entropy require knowledge of the bulk quantum state. In this paper, we derive a novel dual prescription for the generalized entropy that allows us to interpret the leading quantum corrections in a geometric way with minimal input from the bulk state. The equivalence is proven using tools borrowed from convex optimization. The new prescription does not involve bulk surfaces but instead uses a generalized notion of a flow, which allows for possible sources or sinks in the bulk geometry. In its discrete version, our prescription can alternatively be interpreted in terms of a set of Planck-thickness bit threads, which can be either classical or quantum. This interpretation uncovers an aspect of the generalized entropy that admits a neat information-theoretic description, namely, the fact that the quantum corrections can be cast in terms of entanglement distillation of the bulk state. We also prove some general properties of our prescription, including nesting and a quantum version of the max multiflow theorem. These properties are used to verify that our proposal respects known inequalities that a von Neumann entropy must satisfy, including subadditivity and strong subadditivity, as well as to investigate the fate of the holographic monogamy. Finally, using the Iyer-Wald formalism we show that for cases with a local modular Hamiltonian there is always a canonical solution to the program that exploits the property of bulk locality. Combining with previous results by Swingle and Van Raamsdonk, we show that the consistency of this special solution requires the semi-classical Einstein's equations to hold for any consistent perturbative bulk quantum state., Comment: 48 pages, 4 figures v2: minor corrections, a new section on the fate of the MMI inequality, and references updated. v3: matches JHEP version

We study in detail various information theoretic quantities with the intent of distinguishing between different charged sectors in fractionalized states of large-$N$ gauge theories. For concreteness, we focus on a simple holographic $(2+1)$-dimensional strongly coupled electron fluid whose charged states organize themselves into fractionalized and coherent patterns at sufficiently low temperatures. However, we expect that our results are quite generic and applicable to a wide range of systems, including non-holographic. The probes we consider include the entanglement entropy, mutual information, entanglement of purification and the butterfly velocity. The latter turns out to be particularly useful, given the universal connection between momentum and charge diffusion in the vicinity of a black hole horizon. The RT surfaces used to compute the above quantities, though, are largely insensitive to the electric flux in the bulk. To address this deficiency, we propose a generalized entanglement functional that is motivated through the Iyer-Wald formalism, applied to a gravity theory coupled to a $U(1)$ gauge field. We argue that this functional gives rise to a coarse grained measure of entanglement in the boundary theory which is obtained by tracing over (part) of the fractionalized and cohesive charge degrees of freedom. Based on the above, we construct a candidate for an entropic $c$-function that accounts for the existence of bulk charges. We explore some of its general properties and their significance, and discuss how it can be used to efficiently account for charged degrees of freedom across different energy scales., Comment: 30 pages (+4 appendices), 9 figures. v2: typos corrected, references added. v3: General form of the generalized functional highlighted in the main text. Matches published version

We investigate the combined effects of anisotropy and a magnetic field in strongly interacting gauge theories by the gauge/gravity correspondence. Our main motivation is the quark-gluon plasma produced in off-central heavy-ion collisions which exhibits large anisotropy in pressure gradients as well as large external magnetic fields. We explore two different configurations, with the anisotropy either parallel or perpendicular to the magnetic field, focusing on the competition and interplay between the two. A detailed study of the RG flow in the ground state reveals a rich structure where depending on which of the two, anisotropy or magnetic field, is stronger, intermediate geometries with approximate AdS$_4\times \mathbb{R}$ and AdS$_3\times \mathbb{R}^2$ factors arise. This competition is also manifest in the phase structure at finite temperature, specifically in the dependence of the chiral transition temperature on anisotropy and magnetic field, from which we infer the presence of inverse magnetic and anisotropic catalyses of the chiral condensate. Finally, we consider other salient observables in the theory, including the quark-antiquark potential, shear viscosity, entanglement entropy and the butterfly velocity. We demonstrate that they serve as good probes of the theory, in particular, distinguishing between the effects of the magnetic field and anisotropy in the ground and plasma states. We also find that the butterfly velocity, which codifies how fast information propagates in the plasma, exhibits a rich structure as a function of temperature, anisotropy and magnetic field, exceeding the conformal value in certain regimes., Comment: 55 pages, 14 figures, v3: matches published version

In the context of holography, entanglement entropy can be studied either by i) extremal surfaces or ii) bit threads, i.e., divergenceless vector fields with a norm bound set by the Planck length. In this paper we develop a new method for metric reconstruction based on the latter approach and show the advantages over existing ones. We start by studying general linear perturbations around the vacuum state. Generic thread configurations turn out to encode the information about the metric in a highly nonlocal way, however, we show that for boundary regions with a local modular Hamiltonian there is always a canonical choice for the perturbed thread configurations that exploits bulk locality. To do so, we express the bit thread formalism in terms of differential forms so that it becomes manifestly background independent. We show that the Iyer-Wald formalism provides a natural candidate for a canonical local perturbation, which can be used to recast the problem of metric reconstruction in terms of the inversion of a particular linear differential operator. We examine in detail the inversion problem for the case of spherical regions and give explicit expressions for the inverse operator in this case. Going beyond linear order, we argue that the operator that must be inverted naturally increases in order. However, the inversion can be done recursively at different orders in the perturbation. Finally, we comment on an alternative way of reconstructing the metric non-perturbatively by phrasing the inversion problem as a particular optimization problem., Comment: 51 pages, 7 figures and 2 appendices. v2: Matches published version

Quantum corrections to the entanglement entropy of matter fields interacting with dynamical gravity have proven to be very important in the study of the black hole information problem. We consider a one-particle excited state of a massive scalar field infalling in a pure AdS$_3$ geometry and compute these corrections for bulk subregions anchored on the AdS boundary. In the dual CFT$_2$, the state is given by the insertion of a local primary operator and its evolution thereafter. We calculate the area and bulk entanglement entropy corrections at order $\mathcal{O}(N^0)$, both in AdS and its CFT dual. The two calculations match, thus providing a non-trivial check of the FLM formula in a dynamical setting. Further, we observe that the bulk entanglement entropy follows a Page curve. We explain the precise sense in which our setup can be interpreted as a simple model of black hole evaporation and comment on the implications for the information problem., Comment: 52 pages + 2 appendices. Multiple figures. v2: new title, significantly improved introduction and minor changes in the text. v3: extended discussion on interpretation. Version to appear in JHEP

In higher derivative theories, gravity can travel slower or faster than light. With this feature in mind, we revisit the construction of the causal and entanglement wedges in this type of theories, and argue that they must be constructed using the fastest mode instead of null rays. We show that the property of causal wedge inclusion, i.e., the fact that the causal wedge must be contained in the entanglement wedge, leads to more stringent constraints on the couplings than those imposed by hyperbolicity and boundary causality. Our results imply that the full power of subregion-subregion duality could lead to the same conclusions previously obtained based on high energy graviton scattering. We illustrate our findings with a systematic analysis in Gauss-Bonnet gravity., Comment: 21 pages, multiple figures. v3: matches published version

We attempt to construct eternal traversable wormholes connecting two asymptotically AdS regions by introducing a static coupling between their dual CFTs. We prove that there are no semiclassical traversable wormholes with Poincar\'e invariance in the boundary directions in higher than two spacetime dimensions. We critically examine the possibility of evading our result by coupling a large number of bulk fields. Static, traversable wormholes with less symmetry may be possible, and could be constructed using the ingredients we develop here., Comment: 22 pages, 4 figures. v2: minor additions, matches published version

We investigate the effects of anisotropy on the chiral condensate in a holographic model of QCD with a fully backreacted quark sector at vanishing chemical potential. The high temperature deconfined phase is a neutral and anisotropic plasma showing different pressure gradients along different spatial directions, similar to the state produced in noncentral heavy-ion collisions. We find that the chiral transition occurs at a lower temperature in the presence of anisotropy. Equivalently, we find that anisotropy acts destructively on the chiral condensate near the transition temperature. These are precisely the same footprints as the "inverse magnetic catalysis" i.e. the destruction of the condensate with increasing magnetic field observed earlier on the lattice, in effective field theory models and in holography. Based on our findings we suggest, in accordance with the conjecture of [1], that the cause for the inverse magnetic catalysis may be the anisotropy caused by the presence of the magnetic field instead of the charge dynamics created by it. We conclude that the weakening of the chiral condensate due to anisotropy is more general than that due to a magnetic field and we coin the former "inverse anisotropic catalysis". Finally, we observe that any amount of anisotropy changes the IR physics substantially: the geometry is $\text{AdS}_4 \times \mathbb{R}$ up to small corrections, confinement is present only up to a certain scale, and the particles acquire finite widths., Comment: 41 pages, 11 figures

We revisit the recent reformulation of the holographic prescription to compute entanglement entropy in terms of a convex optimization problem, introduced by Freedman and Headrick. According to it, the holographic entanglement entropy associated to a boundary region is given by the maximum flux of a bounded, divergenceless vector field, through the corresponding region. Our work leads to two main results: (i) We present a general algorithm that allows the construction of explicit thread configurations in cases where the minimal surface is known. We illustrate the method with simple examples: spheres and strips in vacuum AdS, and strips in a black brane geometry. Studying more generic bulk metrics, we uncover a sufficient set of conditions on the geometry and matter fields that must hold to be able to use our prescription. (ii) Based on the nesting property of holographic entanglement entropy, we develop a method to construct bit threads that maximize the flux through a given bulk region. As a byproduct, we are able to construct more general thread configurations by combining (i) and (ii) in multiple patches. We apply our methods to study bit threads which simultaneously compute the entanglement entropy and the entanglement of purification of mixed states and comment on their interpretation in terms of entanglement distillation. We also consider the case of disjoint regions for which we can explicitly construct the so-called multi-commodity flows and show that the monogamy property of mutual information can be easily illustrated from our constructions., Comment: 48 pages, multiple figures. v3: matches published version

Holographic theories with classical gravity duals are maximally chaotic: they saturate a set of bounds on the spread of quantum information. In this paper we question whether non-locality can affect such bounds. Specifically, we consider the gravity dual of a prototypical theory with non-local interactions, namely, $\mathcal{N}=4$ non-commutative super Yang Mills. We construct shock waves geometries that correspond to perturbations of the thermofield double state with definite momentum and study several chaos related properties of the theory, including the butterfly velocity, the entanglement velocity, the scrambling time and the maximal Lyapunov exponent. The latter two are unaffected by the non-commutative parameter $\theta$, however, both the butterfly and entanglement velocities increase with the strength of the non-commutativity. This implies that non-local interactions can enhance the effective light-cone for the transfer of quantum information, eluding previously conjectured bounds encountered in the context of local quantum field theory. We comment on a possible limitation on the retrieval of quantum information imposed by non-locality., Comment: 36 pages, 7 figures; v3: matches published version. v4: minor corrections, conclusions unchanged

We present a new family of charged black holes with hyperscaling violating asymptotics and non-trivial horizon topology, for arbitrary Lifshitz exponent $z$ and hyperscaling violation parameter $\theta$. They are $(d+2)$-dimensional analytic solutions to a generalized Einstein-Maxwell-Dilaton theory with an additional vector field supporting spherical or hyperbolic horizons. The hyperbolic solutions are only consistent with the null energy condition for $1 \le z < 2$ and $\theta = d(z-1)$, which explains why they are absent in the pure Lifshitz case. We further study the (extended) thermodynamics of these black holes, both in the fixed charge and fixed potential ensemble, and find that phase transitions only occur for spherical black holes with $1 \le z \le 2$ and any $\theta$., Comment: 38 pages, multiple figures, 1 appendix. v2: minor additions and references added. v3: version published in CQG: https://dx.doi.org/10.1088/1361-6382/ab0094

In the holographic correspondence, subregion duality posits that knowledge of the mixed state of a finite spacelike region of the boundary theory allows full reconstruction of a specific region of the bulk, known as the entanglement wedge. This statement has been proven for local bulk operators. In this paper, specializing first for simplicity to a Rindler wedge of AdS$_3$, we find that generic curves within the wedge are in fact not fully reconstructible with entanglement entropies in the corresponding boundary region, even after using the most general variant of hole-ography, which was recently shown to suffice for reconstruction of arbitrary spacelike curves in the Poincare patch. This limitation is an analog of the familiar phenomenon of entanglement shadows, which we call 'entanglement shade'. We overcome it by showing that the information about the nonreconstructible curve segments is encoded in a slight generalization of the concept of entanglement of purification, whose holographic dual has been discussed very recently. We introduce the notion of 'differential purification', and demonstrate that, in combination with differential entropy, it enables the complete reconstruction of all spacelike curves within an arbitrary entanglement wedge in any 3-dimensional bulk geometry., Comment: 30 pages, 5 figures. v2: references added, typos corrected. v3: published version

Holographic theories with classical gravity duals are maximally chaotic; i.e., they saturate the universal bound on the rate of growth of chaos. It is interesting to ask whether this property is true only for leading large $N$ correlators or if it can show up elsewhere. In this Letter we consider the simplest setup to tackle this question: a Brownian particle coupled to a thermal ensemble. We find that the four-point out-of-time-order correlator that diagnoses chaos initially grows at an exponential rate that saturates the chaos bound, i.e., with a Lyapunov exponent $\lambda_L=2\pi/\beta$. However, the scrambling time is parametrically smaller than for plasma excitations, $t_*\sim\beta \log \sqrt{\lambda}$ instead of $t_*\sim\beta \log N^2$. Our result shows that, at least in certain cases, maximal chaos can be attained in the probe sector without the explicit need of gravitational degrees of freedom., Comment: v3: minor additions and typos corrected. Version to appear in PRL

We initiate a non-perturbative study of anisotropic, non-conformal and confining gauge theories that are holographically realized in gravity by generic Einstein-Axion-Dilaton systems. In the vacuum our solutions describe RG flows from a conformal field theory in the UV to generic scaling solutions in the IR with generic hyperscaling violation and dynamical exponents $\theta$ and $z$. We formulate a generalization of the holographic c-theorem to the anisotropic case. At finite temperature, we discover that the anisotropic deformation reduces the confinement-deconfinement phase transition temperature suggesting a possible alternative explanation of inverse magnetic catalysis solely based on anisotropy. We also study transport and diffusion properties in anisotropic theories and observe in particular that the butterfly velocity that characterizes both diffusion and growth of chaos transverse to the anisotropic direction, saturates a constant value in the IR which can exceed the bound given by the conformal value., Comment: 6 pages, 4 figures; v2: minor improvements, references added, version accepted for publication in PRL

In the context of the AdS/CFT correspondence, we study bulk reconstruction of the Poincare wedge of AdS$_3$ via hole-ography, i.e., in terms of differential entropy of the dual CFT$_2$. Previous work had considered the reconstruction of closed or open spacelike curves in global AdS, and of infinitely extended spacelike curves in Poincare AdS that are subject to a periodicity condition at infinity. Working first at constant time, we find that a closed curve in Poincare is described in the CFT by a family of intervals that covers the spatial axis at least twice. We also show how to reconstruct open curves, points and distances, and obtain a CFT action whose extremization leads to bulk points. We then generalize all of these results to the case of curves that vary in time, and discover that generic curves have segments that cannot be reconstructed using the standard hole-ographic construction. This happens because, for the nonreconstructible segments, the tangent geodesics fail to be fully contained within the Poincare wedge. We show that a previously discovered variant of the hole-ographic method allows us to overcome this challenge, by reorienting the geodesics touching the bulk curve to ensure that they all remain within the wedge. Our conclusion is that all spacelike curves in Poincare AdS can be completely reconstructed with CFT data, and each curve has in fact an infinite number of representations within the CFT., Comment: 37+3 pages, multiple figures. v2: typos corrected, matches published version

In large-$N_c$ conformal field theories with classical holographic duals, inverse coupling constant corrections are obtained by considering higher-derivative terms in the corresponding gravity theory. In this work, we use type IIB supergravity and bottom-up Gauss-Bonnet gravity to study the dynamics of boost-invariant Bjorken hydrodynamics at finite coupling. We analyze the time-dependent decay properties of non-local observables (scalar two-point functions and Wilson loops) probing the different models of Bjorken flow and show that they can be expressed generically in terms of a few field theory parameters. In addition, our computations provide an analytically quantifiable probe of the coupling-dependent validity of hydrodynamics at early times in a simple model of heavy-ion collisions, which is an observable closely analogous to the hydrodynamization time of a quark-gluon plasma. We find that to third order in the hydrodynamic expansion, the convergence of hydrodynamics is improved and that generically, as expected from field theory considerations and recent holographic results, the applicability of hydrodynamics is delayed as the field theory coupling decreases., Comment: 37+5 pages, multiple figures. v2: typos corrected, matches published version